Degenerate Reflections in Acoustics of Solids. II: Orthorhombic Crystals
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ICAL PROPERTIES OF CRYSTALS
Degenerate Reflections in Acoustics of Solids. II: Orthorhombic Crystals V. N. Lyubimova,* a
Shubnikov Institute of Crystallography, Federal Scientific Research Centre “Crystallography and Photonics,” Russian Academy of Sciences, Moscow, 119333 Russia *e-mail: [email protected] Received March 1, 2018; revised March 1, 2018; accepted March 15, 2018
Abstract—The conditions under which a bulk acoustic wave, incident on a crystal at a certain angle and reflected from the crystal boundary with vacuum, generates two or one reflected bulk waves rather than three (as in the general case) are analyzed. These degenerate reflections are considered using the example of orthorhombic crystals. DOI: 10.1134/S1063774519030180
INTRODUCTION When a bulk acoustic wave propagating in a crystal reflects off from its boundary with vacuum, several reflected waves are formed. The reflected waves arise in the following combinations: three bulk waves, two bulk waves accompanied by one localized at the boundary, and one bulk wave with two localized waves. Note that only bulk partial waves are reflected ones. This becomes clear if we recall that the physical image of a plane wave is realized in the form of an acoustic beam with a width significantly exceeding the wavelength. Therefore, a localized partial wave is present only in the vicinity of acoustic “spot” on the surface, where it provides the fulfillment of boundary conditions for the entire superposition of waves and is not directly related to reflection (i.e., energy removal from the boundary). Hence, there must be at least one bulk reflected wave among partial waves. The realizability of a specific version with a certain number of reflected waves depends on the symmetry of crystal, its elastic anisotropy, the propagation geometry (the boundary orientation and the propagation direction of the incident wave), and the acoustic branch the incident wave belongs to. At certain combinations of these factors, the amplitude of one of the reflected bulk waves may be zero. The reflection turns out to be degenerate, i.e.,there may be several versions for reflected waves: two bulk waves; one bulk wave and one localized wave; and, finally, one bulk wave. These reflections were considered in cubic [1, 2] and hexagonal [3] crystals. The versions in which the incident wave generates a reflected bulk wave close to the eigenmode (peculiar bulk wave) have also been studied. When the reflected-wave propagation direction is
close to the crystal surface, all energy of the incident wave can be concentrated in a narrow reflected beam, which is accompanied by only a single wave localized at the surface. This reflection is degenerate [4–6]. Irrelative of resonances, the general theory of reflections in crystals of arbitrary symmetry was developed in [7–11]. There are degenerate reflections in isotropic media as well [12–15]; these reflections were described from the general point of view in [16]. In this work, degenerate reflections in anisotropic crystals are considered. The genera
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